inequalities

**ose90** · April 3rd 2009, 08:27 PM

given that x+y=1, prove that 1/x + 1/y >= 4

Appreciate any help/hints!

**NonCommAlg** · April 3rd 2009, 09:08 PM

Originally Posted by ose90

given that x+y=1, prove that 1/x + 1/y >= 4

Appreciate any help/hints!

you also need to have x > 0 and y > 0. first note that from $(t-1)^2 \geq 0,$ we get: $t + \frac{1}{t} \geq 2,$ for any $t > 0.$ thus: $\frac{1}{x} + \frac{1}{y} = (x+y) \left(\frac{1}{x} + \frac{1}{y} \right)=2 + \frac{x}{y} + \frac{y}{x} \geq 4.$

**ose90** · April 5th 2009, 01:51 AM

Originally Posted by NonCommAlg

you also need to have x > 0 and y > 0. first note that from $(t-1)^2 \geq 0,$ we get: $t + \frac{1}{t} \geq 2,$ for any $t > 0.$ thus: $\frac{1}{x} + \frac{1}{y} = (x+y) \left(\frac{1}{x} + \frac{1}{y} \right)=2 + \frac{x}{y} + \frac{y}{x} \geq 4.$

Thanks alot!

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